EVALUACIÓN DE PROGRAMAS DE ALFABETIZACIÓN DE ADULTOS
4. Personal que desarrolla los programas
1. The conditions are those necessary for two-fund separation, namely (in addition to those already assumed)
• homogeneous expectations
• asset markets frictionless with costless information simultaneously available to all individuals
• all assets marketable and perfectly divisible
• no market imperfections such as taxes or restrictions on selling short 2.
(f) The required rate of return on Donovan Company is
The expected return, E(R )*j = 15% (see c above). Because Donovan earns more than is required, we might expect its price to increase so that in equilibrium its expected rate of return equals 14.6%.
3. Now we are dealing with sample data; therefore, we can use statistics formulae for sample mean and variance.
Note that the sample variance is calculated by dividing the sum of mean deviations squared by N – 1, the number of observations minus one. We subtract one because we lose one degree of freedom in estimating the mean. Covariance calculations also are divided by N – 1.
− −
62 Copeland/Shastri/Weston • Financial Theory and Corporate Policy, Fourth Edition
The market requires 12.9% and the expected (i.e., anticipated) rate of return is only 10%.
Therefore, we would expect the Milliken company to decline in value.
4. See table S6.1, Panel A.
6. Other than the usual perfect market assumptions, and given that we have rational, risk-averse, expected-end-of-period-utility-of-wealth maximizers, the key assumptions are that 1) investors have homogeneous expectations and 2) all assets are perfectly divisible and marketable. It is not necessary that there be risk-free assets. Given the two above assumptions, all individuals will perceive the same efficient set and all assets will be held in equilibrium. If every individual holds an efficient portfolio, and all assets are held, then the market
Table S 6.1
Panel A: Estimates of Market Parameters Year
Panel B: Calculation of Beta for General Motors Year
64 Copeland/Shastri/Weston • Financial Theory and Corporate Policy, Fourth Edition
portfolio must also be efficient because it is merely the sum of all efficient portfolios held by all individuals.
7. Yes. Given a riskless asset, two-fund separation obtains, and if you can observe any one of the following: 1) the percent of the investor’s portfolio held in the market portfolio, 2) the β of the investor’s portfolio, or 3) the expected return on the investor’s portfolio, you can tell how risk averse he is. These three measures are equivalent. To see this, note that
E(Rp) = Rf+ [E(Rm) – Rf] βp
where a is the percent invested in the market portfolio. Therefore, a = βp. Finally, β = β − + βp f(1 a) ma
where βf= 0 is the beta of the risk-free asset. Therefore, βp= aβm
and since βm= 1.0
βp= a
The greater the percentage, a, of the market portfolio held in the investor’s total portfolio, the less risk averse the investor is. This result does not change in a world without a risk-free asset because the zero-β portfolio may be substituted for the risk-free asset without changing any results.
8. Systematic risk is defined as market risk, i.e., a portfolio’s variance that can be explained in terms of market variance. With risk-free borrowing and lending, all efficient portfolios are linear combinations of the risk-free asset and the market portfolio. The capital market line is made up of a linear efficient set as in Figure 6.2. Because the risk-free asset has no variance, total variance in every efficient portfolio is contributed solely by the market portfolio. Thus, all risk is systematic.
Algebraically, the variance in any efficient portfolio would be
+ − = σ + − σ + − σ
where a represents the percentage of the efficient portfolio invested in the market portfolio. The percentage, a, is equal to the portfolio’s βp, as proven in problem 6.7. β is the measure of systematic risk.
Thus, σ = β σp p m.
9. E(R ) Rj − f+[E(R ) R ] m − f βj
− = β =
− j
.2 .05 .15 .05 1.5
We know that efficient portfolios have no unsystematic risk. The total risk (equation 6.12) is σ = β σ + σ2j 2j 2m ε2
and since the unsystematic risk of an efficient portfolio, σ , is zero, 2ε σ = βj j ms =1.5(.2)=.3 or 30%, σ =2j .09 or 9%
The definition of correlation (from Chapter 5, equation 17) is
σ σ
Since rjm= 1.0, we again have the result that efficient portfolios are perfectly correlated with the market (and with each other).
10. Assuming equilibrium, all assets must be priced so that they fall along the security market line.
Therefore, we can find the βk for Rapid Rolling. βk is its systematic risk.
Total risk is defined by equation 6.12. We can use it to find Rapid Rolling’s unsystematic risk, σ 2ε
ε
The CAPM assumes that the market is in equilibrium and that investors hold efficient portfolios, i.e., that all portfolios lie on the security market line.
66 Copeland/Shastri/Weston • Financial Theory and Corporate Policy, Fourth Edition
(b) Let “a” be the percent invested in the risk-free asset. Portfolio return is the point on the security market line where
Therefore, (1 – a) = 50%, i.e., the individual should put 50% of his portfolio into the market portfolio.
Figure S6.1 (Problem 6.11) The security market line
12. Assuming that the company pays no dividends, the one period rate of return, Rj, is
= 1 − 0
Substituting in the appropriate numbers and solving for P0, we have
− = + −
13. (a) A zero-beta portfolio has zero covariance with the market portfolio, by definition. Also, using matrix notation, the covariance between two portfolios is
cov=W1'ΣW 2
where
'
W 1 = the row vector of weights in the zero -β portfolio Σ = the variance-covariance matrix of two risky assets W2= the column vector of weights in the index portfolio By setting cov = 0, we can solve for W . 1'
In order to have two equations and two unknowns, we also use the fact that the weights must always sum to one, i.e.,
X1+ X2= 1 Solving this system of equations we have
1 2
This implies putting 278 percent of the portfolio wealth into asset 2 and –178 percent into asset 1.
Therefore, the expected return on the zero-β portfolio is E(R )p 1.78E( ) 2.78E( )
(b) The vector of weights in the minimum variance portfolio can be found by using equation 5.21.
xy x y
From the variance-covariance matrix, Σ we know that , σ =2x .01 cov( ,X Y)=0
Thus, the minimum variance portfolio consists of 39 percent in asset 1 and 61 percent in asset 2.
68 Copeland/Shastri/Weston • Financial Theory and Corporate Policy, Fourth Edition
(c) Using the covariance definition again, we have cov = W1' WΣ 2 E(RI) = the expected return on the efficient index portfolio
βjI= the covariance between the returns on the jth asset and the index portfolio, standardized by the variance of the index portfolio
Substituting in the estimated parameters, we have
j jI
14. Using matrix notation, the definition of covariance is cov(X, Y)=Wx' Σ Wy
where
'
W x = a row vector of weights in portfolio A Wy= a column vector of weights in portfolio B
Σ = the variance-covariance matrix of A and B Substituting in the appropriate numbers, we have
.01 .02 .6 15. Using the definition of the correlation coefficient, we have
σ σ
k,m
k m
cov(k, m) r =
Substituting the correct numbers and solving for cov(k, m), cov( ) .8= (.25)(.2)k, m
cov(k, m) = .8(.25)(.2) = .04
Using the definition of β we can calculate the systematic risk of Knowlode
β = = =
The systematic risk of a portfolio is a weighted average of the asset’s β ’s. If “a” is the percent of Knowlode,
βp= (1 – a)βf+ aβk
1.6 = (1 – a) (0) + a(1.0) a = 1.6 or 160%
In this case the investor would borrow an amount equal to 60 percent of his wealth and invest 160 percent of his wealth in Knowlode in order to obtain a portfolio with a β of 1.6. (This analysis assumes investors may borrow at the risk-free rate.)
16. First, we need to know E(Rm) and Rf. Since each of the assets must fall on the security market line in equilibrium, we can use the CAPM and E(Ri).
Asset 1: E(Ri) = Rf+ [E(Rm) – Rf] βi
7.6 = Rf+ [E(Rm) – Rf].2 Asset 2: 12.4 = Rf+ [E(Rm) – Rf] .8
This gives two equations with two unknowns. Multiplying the first equation by 4 and subtracting the second equation, we have
4(7.6) = 4Rf+ [E(Rm) – Rf] .8
Substituting the value of Rf into equation 1, we have
7.6 = 6 + [E(Rm) – 6] .2 should also fall on the security market line.
For asset 3, we have
E(R3) = 6 + [14 – 6]1.2 = 15.6
70 Copeland/Shastri/Weston • Financial Theory and Corporate Policy, Fourth Edition
and for asset 4 we have
E(R4) = 6 + [14 – 6] 1.6 = 18.8 The expected rate of return and β for the current portfolio are
5
In order to achieve a new portfolio with a 12 percent expected rate of return, we subtract X percent from our holdings in the risk-free asset and add X percent to new holdings in the market portfolio.
12 = E(Rp) = .1(7.6) + .1(12.4) + .1(15.6) + .2(18.8)
Therefore, the new portfolio holdings will be
Return on Asset i βi Percent in Asset i
The expected return of the new portfolio is 12 percent and its β is .75.
If you hold only the risk-free asset and the market portfolio, the set of weights that would yield a 12 percent expected return can be determined as follows:
W1= amount in risk-free asset W2= amount in market portfolio
.06W1+ .14W2= .12 W1+ W2= 1
We have two equations with two unknowns. Multiplying the second equation by .06 and subtracting from the first gives
+ =
A portfolio with 25 percent in the risk-free asset and 75 percent in the market portfolio has an expected return of 12 percent.
17. We know from the CAPM
E(Rj) = Rf+ [E(Rm) – Rf]βj
Substituting the correct numbers for βj
.13 = .07 + (.08)βj
If the rate of return covariance with the market portfolio doubles, βj will also double since β = σ
so the new required return for the security will be
Required Rj= .07 + (.08)1.5 = .19 = 19%
At the original rate of return, 13 percent, and given a present price of P0= $40, the expected future price, P1, was
P0(1 + r) = P1
40(1.13) = $45.20
If the expected future price is still $45.20 but the covariance with the market has doubled, so that the required return is 19 percent, then the present price will be
P0= P1(1 + r)–1= 45.20/1.19
= $37.98
72 Copeland/Shastri/Weston • Financial Theory and Corporate Policy, Fourth Edition
Since the expected rate of return is greater than the required rate of return, investment in the security would be advisable.
To find the percentage change in the security’s price that would result in a return of 10 percent instead of 12 percent, solve for P* in terms of P0:
If the security’s price increases more than 1.82 percent, reverse the investment decision.
(b) There are two possibilities for the low ex post return. First, the expected rate of return and the estimated risk could have been overestimated. The second possibility is that after the fact, the market unexpectedly fell. The ex post market rate of return which would have resulted in a 5 percent rate of return for the security in question is
= + − β The ex post security market line is depicted in Figure S6.2.
Figure S6.2 The ex post security market line
19. (a) Given that some assets are nonmarketable, the equilibrium pricing equation is given by Mayers (1972) in equation 6.26.
j f m
E(R )=R + λ[V cov(R ,Rj m)+cov(R ,Rj H)]
This implies that the individual will take into consideration not only the covariance of an asset with the market portfolio, but also its covariance with the portfolio of nonmarketable assets. This may be interpreted as a form of three-fund separation, where the three funds are 1) the risk-free asset, 2) the portfolio of risky marketable assets, and 3) the portfolio of risky nonmarketable assets. Separation is still valid in the sense that the marginal rate of substitution between risk and return is independent of individuals’ utility preferences.
(b) If the risk-free rate is nonstochastic, then we can use Merton’s (1973) continuous time model as shown in equation 6.28
j f 1 m f 2 N f
E(R )= + γr [E(R )−r ]+ γ [E(R )−r ]
Once again, three-fund separation obtains. Every investor will hold one of three funds: 1) the risk-free asset, 2) the market portfolio, and 3) a hedge portfolio chosen to hedge against unforeseen changes in the future risk-free rate. We have separation because the market price of risk (the marginal rate of substitution between risk and return) is independent of individuals’ utility functions.
20. (a) The covariance between investor I’s index and asset A is
The variance of investor I’s index is
[ ]
Repeating the exercise for investor J we have
74 Copeland/Shastri/Weston • Financial Theory and Corporate Policy, Fourth Edition
(b) Both investors will require the same rate of return on asset A. They must. They have homogenous expectations therefore they perceive the same risk-return combination. The expected return on asset A is 30%.
(c) The key to understanding this problem is Roll’s critique. The two investors choose different index portfolios, but both portfolios are efficient. They must be because this (over-simplified) world has only two risky assets. Any combination of them will lie on the mean variance opportunity set.
Consequently, even though the investors perceive different security market lines and different betas, they will perceive the same expected rate of return on any portfolio. In order to numerically demonstrate this result the zero-beta portfolios and security market lines for each investor are computed below.
The zero-beta portfolio is the minimum variance portfolio which has zero covariance with the index portfolio. Since all portfolios lie on the minimum variance opportunity set in this two-asset world, we need be concerned only with the zero covariance condition. If we define Rz as the return on the zero-beta portfolio and Wz as its vector of weights, then we require that
COV(R ,Rz I)=Wz'ΣWI=0 The appropriate numbers for investor I are:
Solving, we have one equation and two unknowns
+ =
z1 z2
.75(.0081)W .25(.0025)W 0 However, we also know that the weights must add to one
+ =
z1 z2
W W 1
This gives two equations and two unknowns, as shown below
+ = Thus, for investor I the expected return on his zero-beta portfolio is
= +
Therefore, the security market line perceived by investor I can be written as follows:
j z I z I
Recalling that investor I estimated a β of 1.289 for asset A, we can now verify that investor I expects a 30% rate of return on asset A.
= + −
=
E(R )A .1885 [.2750 .1885]1.289 30%
Next, we repeat the same procedure for investor J. The only difference is that investor J uses a different index portfolio. Investor J’s zero-beta portfolio will be
'
The two simultaneous equations are
+ =
and their solution is
z1= −
W .4464, Wz2 =1.4464 The expected rate of return on investor J’s zero-beta portfolio is
( z)= −.4464(30%) 1.4464(20%) 15.54%+ = E R
Investor J’s security market line equation is
j j
E(R )=.1554 [.2500 .1554] ,+ − β SML for J.
Finally, recalling that investor J estimated a β of 1.528 for asset A, we see that his expected return on asset A is
= + −
=
E(R )A .1554 [.2500 .1554]1.528 30%
This numerical example demonstrates that although investors I and J estimate different betas, they will have the same expected return for asset A.
21. (a) Using the CAPM, the expected return on her portfolio is
i f M f i
Using the fact that E(Ri) = 14.2% for her portfolio and using the APT equation, we have
= = + β + β
76 Copeland/Shastri/Weston • Financial Theory and Corporate Policy, Fourth Edition
(b) Writing the APT equations once more, we have
= = + β + β
i i1 i2
E(R ) .142 .08 (.05) (.11) Substituting in the value of bi2 as zero, we have
= + β
− = β
i1
i1
.142 .08 .05 .142 .08
.05
= βi1
1.24